Multiplication and Division of Integers

Multiplication and division of integers can be done by following a specific set of rules that make it easy to handle positive and negative integers. Let us take a look at the rules for multiplying and dividing integers:

Multiplication 

• If two integers with the same sign are multiplied, then the product will always have a positive sign. 

• If two integers with different signs are multiplied, then the product will always have a negative sign.

• Any number multiplied by zero is always zero. 

Division 

• Dividing two integers with the same sign gives a quotient with a positive sign. 

• Dividing integers with different signs results in a negative quotient. 

• Division by zero is undefined.

It is important to know the properties of multiplication and division of integers so that it becomes easier to solve problems involving integers. The properties are mentioned below: 

Closure Property: Closure property states that the product of any two integers is also an integer.

Commutative Property: In this property, the order of multiplication does not affect the result.

Associative Property: In this property of multiplication, the result remains unchanged even when the grouping of integers is changed.

Multiplicative Identity Property: Multiplying 1 by any integer will result in the integer itself.

Multiplication by Zero Property: Multiplying any integer by zero results in zero.

Distributive Property: The distributive property states that multiplication distributes over addition and subtraction.

Given below are the properties of the division of integers:

Non-Commutative Property: If the order of division is changed, then the result also changes.

Non-Associative Property: If the grouping of integers is changed in division, then the result also changes. 

Division by 1 Property: When an integer is divided by 1, the result is the integer itself. 

Division by Itself Property: Dividing any non-zero integer by itself will always result in 1.

Division by Zero is Undefined: If any number is divided by zero, then it is undefined.
 

Multiplication of integers follows specific sign rules. When multiplying two integers with the same sign (both positive or both negative), the product is positive. If integers have different signs, the product is negative. If any number is multiplied by 0, then the result will be 0. 

How to Solve Multiplication of Integers? 

To solve multiplication of integers, take a look at the steps given below:

Step 1: Identify the Signs of the Numbers
First, we have to check if the integers being multiplied are positive or negative.
Use the sign rules for multiplication:
Positive × Positive = Positive
Negative × Negative = Positive
Positive × Negative = Negative
Negative × Positive = Negative

Step 2: Multiply the Absolute Values
Ignore the signs and multiply the numbers normally.

Step 3: Apply the Correct Signs
If both numbers have the same sign, the product is positive.
If the numbers have different signs, then the product is negative.

Step 4: Write the Final Answer 
Ensure that the final answer includes the correct sign. To verify the result, reverse the operation (division).
 

Division of integers follows specific rules based on the signs of the numbers. When dividing two integers with the same sign (both positive or both negative), the quotient is positive. When dividing two integers with different signs (one positive and one negative), the quotient is negative. If zero is divided by any integer, then the result is always zero. If an integer is divided by 0, then the result is undefined. 

How to Solve Division of Integers?

 
Take a look at the following steps to solve the division of integers:

Step 1: Identify the Signs
Check if the divisor and dividend have positive or negative signs. Once we ascertain the signs, we can apply the sign rules for division, which are given below: 
Positive ÷ Positive = Positive
Negative ÷ Negative = Positive
Positive ÷ Negative = Negative
Negative ÷ Positive = Negative

Step 2: Divide the Absolute Values

While dividing the absolute values, we can ignore the signs and simply continue the division process.

Step 3: Apply the Correct Sign
Now it is time to identify the sign of the quotient. One should follow the sign rules mentioned in step 1.

Step 4: Write the Final Answer 
After writing the answer, cross-verify it by multiplying the quotient by the divisor. If the product is the same as the dividend, then the answer is correct.

The multiplication and division of integers have numerous applications across various fields. Let us explore how the multiplication and division of integers are used in different areas:

Business:
Multiplication and division of integers are essential to run a business, as they can be used to calculate the profits and losses.

Temperature Changes:
Integer multiplication and division are used in weather forecasting and to study climate change. If the temperature drops with every passing hour by 2°C, then the total decrease in temperature over 6 hours can be calculated as −2 × 6 = −12°C. 

Elevation and Depth Calculations:
The multiplication and division of integers play an important role in aviation when calculating elevations and depths. It is also used in submarines to calculate the depth of the ocean.